Maximal Feeding with Active Prey-Switching: a Kill-The-Winner Functional Response and Its Effect on Global Diversity and Biogeography ⇑ S.M

Progress in Oceanography 120 (2014) 93–109

Contents lists available at ScienceDirect

Progress in Oceanography

journal homepage: www.elsevier.com/locate/pocean

Maximal feeding with active prey-switching: A kill-the-winner functional response and its effect on global diversity and biogeography ⇑ S.M. Vallina a,b, , B.A. Ward a,1, S. Dutkiewicz a, M.J. Follows a a Earth, Atmospheric and Planetary Sciences, MIT, Cambridge, USA b Marine Sciences Institute, ICM – CSIC, Barcelona, Catalonia, Spain article info abstract

Article history: Predators’ switching towards the most abundant prey is a mechanism that stabilizes population dynam- Received 1 June 2012 ics and helps overcome competitive exclusion of species in food webs. Current formulations of active Received in revised form 8 August 2013 prey-switching, however, display non-maximal feeding in which the predators’ total ingestion decays Accepted 9 August 2013 exponentially with the number prey species (i.e. the diet breadth) even though the total prey biomass Available online 5 September 2013 stays constant. We analyse three previously published multi-species functional responses which have either active switching or maximal feeding, but not both. We identify the cause of this apparent incom- Keywords: patibility and describe a kill-the-winner formulation that combines active switching with maximal feed- Zooplankton grazing ing. Active switching is shown to be a community response in which some predators become prey- Foodweb stability Ecosystem diversity selective and the formulations with maximal or non-maximal feeding are implicitly assuming different Ocean modeling food web configurations. Global simulations using a marine ecosystem model with 64 phytoplankton species belonging to 4 major functional groups show that the species richness and biogeography of phytoplankton are very sensitive to the choice of the functional response for grazing. The phytoplankton biogeography reflects the balance between the competitive abilities for nutrient uptake and the degree of apparent competition which occurs indirectly between species that share a common predator species. The phytoplankton diversity significantly increases when active switching is combined with maximal feeding through predator-mediated coexistence. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction 1973). Active switching makes the proportion of a given prey attacked to change from less than expected to more than expected Active prey-switching is a predatory behavior that has been as the relative abundance of that prey increases (Hassell, 2000). documented in natural ecosystems (Murdoch, 1969, 1975; Hughes From an ecosystem modeling perspective, active switching is an and Croy, 1993; Kiørboe et al., 1996; Gismervik and Andersen, interesting property because it allows for a greater degree of species 1997; Elliott, 2006; Kempf et al., 2008; Kiørboe, 2008; Kalinkat co-existence in competitive food webs (Vallina and Le Quéré, 2011; et al., 2011) and is known to stabilize ecosystem dynamics Prowe et al., 2012a,b). Multi-species ecosystem models can overcome (Murdoch and Oaten, 1975; Haydon, 1994; Armstrong, 1999; the competitive exclusion principle (Hardin, 1960; Hutchinson, 1961; Morozov, 2010). Active switching differs from passive switching Armstrong and McGehee, 1980) by including some form of active in that the predators’ switching is variable and based on relative switching (Adjou et al., 2012). In a broad sense, selective predation prey density (i.e. frequency-dependent predation), rather than can be argued to fit within the ‘‘killing the winner’’ theory, which is being fixed and based on constant prey preferences (see sometimes invoked to explain the high diversity we observe in micro- Gentleman et al. (2003) for a review). Thus, active switching repre- bial communities (Thingstad and Lignell, 1997; Thingstad, 2000). The sents a behavioral change of the predator (Gentleman et al., 2003), basic idea is that the most abundant bacteria types will be killed either in terms of feeding strategy (e.g. from passive suspension preferentially by host-selective viral lysis. Therefore, the coexistence feeding to active ambush feeding) (Kiørboe et al., 1996; Gismervik of competing bacterial species is ensured by the presence of viruses and Andersen, 1997; Wirtz, 2012b) or learning how to increase the that kill-the-winner, whereas the differences in substrate affinity efficiency of capturing and handling certain prey types (Murdoch, between the coexisting bacterial species determine viral abundance (Thingstad, 2000). Active switching follows conceptually the same principle but for predator–prey selectivity. ⇑ Corresponding author at: Marine Sciences Institute, ICM – CSIC, Barcelona, However, current formulations of active prey-switching show Catalonia, Spain. Tel.: +34 932 309 607; fax: +34 932 309 555. anomalous dynamics, like antagonistic feeding and sub-optimal E-mail address: [email protected] (S.M. Vallina). 1 Current address: CERES-ERTI, École Normale Supérieure, Paris, France. feeding in which predators are unable to maximize the ingestion

0079-6611/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.pocean.2013.08.001 94 S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109 of the total food available when it becomes divided among several term ‘‘species’’ in a very broad and general sense, simply denoting prey (Tilman, 1982; Holt, 1983; Gentleman et al., 2003). In antag- variability of the phytoplankton traits for nutrient uptake. An alterna- onistic feeding, if total food abundance is evenly distributed among tive term could be phytoplankton ecotypes (Dutkiewicz et al., 2009). many prey, it will give a smaller total ingestion than if the same total food is concentrated in one prey species (Tilman, 1982). In other 2. Functional responses words, for a given total food availability, the most even distribution of prey biomass will give the lowest total ingestion. Sub-optimal The functional response describes how the ingestion rate of a feeding occurs when an increase in the abundance of one prey predator changes with prey density. That is, it gives the function can also result in a decrease of ingestion, despite that total food that relates the amount of prey ingested per predator and unit of is actually increasing. Sub-optimal feeding is an extreme form of time to the density of the prey in the environment (Murdoch, antagonistic feeding (Gentleman et al., 2003). When taken to the 1973). Although there are many functional responses described limit where each prey contributes to an infinitesimal fraction of in the literature (Gentleman et al., 2003), the most common are the total prey abundance, these two modes of non-maximal feed- the Holling Type I, II, III (Holling, 1959) and the Ivlev (Ivlev, ing imply that the total ingestion by the predators will tend to- 1961) functions for single prey type ingestion (see Fig. 1). A pred- wards zero, even if the combined biomass of all their prey is high. ator can theoretically change mode between functional responses These formulation inconsistencies are conceptually problematic (Real, 1977, 1979; Wirtz, 2012a). This work will only focus on tran- and have been used to warn against the use of active switching sitions between the Type II (hyperbolic) and Type III (sigmoidal) functional responses in ecosystem models (Gentleman et al., responses. The Type II response gives a decelerating ingestion rate 2003). Here we argue that the problem does not lie with the use with increasing prey density, and thus provides prey safety only at of active switching per-se but with the fact that current formula- high abundances. This leads to a lower per capita risk of being ea- tions are not completely satisfactory representations of switching ten at high prey densities and to a higher per capita risk of being behavior (Holt, 1983; Mitra and Flynn, 2006; Anderson et al., eaten at low prey densities, which tends to destabilize predator– 2010). Total ingestion should ideally depend on the total food prey interactions. The Type III response, on the other hand, also amount and its quality but not necessarily on the biomass distribu- provides prey safety at low abundances, which tends to stabilize tion of the prey. In such a functional response all the prey would be predator–prey interactions. Type III responses can be explained perfectly substitutable for equal fixed preferences (Tilman, 1982) in terms of optimal foraging theory as a way to optimize the energy and feeding will always be maximal. The original Holling Type II intake from feeding with respect to the energy cost of foraging functional response is probably the best known example (Holling, (MacArthur and Pianka, 1966; Pahlow and Prowe, 2010). 1959; Gentleman et al., 2003). However, it does not allow for active Most multi-species functional responses for predation are sim- prey-switching and therefore the competitive exclusion among the ply variations of the original Holling Type II/III formulations for one prey is very difficult to prevent and the ecosystem stability is dras- prey (Holling, 1959) but extended to consider many prey (Mur- tically reduced (Gismervik and Andersen, 1997). Ward et al. (2012) doch, 1973), and can be found in the literature as several mathe- suggests an equation for the switching between herbivory and car- matically equivalent equations. The general expression common nivory. We use a similar approach for the switching between indi- to all four functional responses in this study is: vidual prey. The first objective of this work is to identify the origin of the ob- Gj ¼ gmaxZdjQ ¼ V maxdjQ ð1Þ served incompatibility between active switching and maximal X X G G V Q d V Q 2 ingestion in current formulations of predation on multiple prey ¼ j ¼ max j ¼ max ð Þ (Gentleman et al., 2003). We evaluate three classical formulations of predation: two that exhibit switching but non-maximal inges- 1.2 tion (one sub-optimal, one antagonistic); and one that exhibits Type I maximal ingestion but no-switching. We also describe a kill-the- ] 1 Type II winner (KTW) functional response that combines active switching −1 with maximal ingestion (see Appendix A). Maximal and non-max- d Type III −3 imal ingestion are shown to arise from the implicit assumptions of 0.8 Ivlev the food web configuration inherent to each functional response (see Appendix B). 0.6 The second objective of this work is to evaluate how grazing functional responses affect the simulated global distributions of 0.4 marine phytoplankton diversity and biogeography. The choice of the grazing response has already been shown to drastically change the simulated distributions of phytoplankton biogeography Grazing rate [mmol m 0.2 (Anderson et al., 2010) and diversity (Prowe et al., 2012a). How- ever, these results were obtained from comparing ‘‘passive-switch- 0 ing with maximal feeding’’ formulations (i.e. Real’s) against 0 20 40 60 80 100 ‘‘active-switching with non-maximal feeding’’ formulations (i.e. Prey biomass [mmol m−3] Fasham’s and Ryabchenko’s). Following a similar approach here we also evaluate the effect of the new KTW formulation that com- Fig. 1. Shape of the classical Holling Type (I, II, III) and Ivlev functional responses for ingestion [mmol m3 d1] upon a single prey type [mmol m3] with a maximum bines active-switching with maximal feeding. Thus we imple- grazing rate of 1.0 [mmol m3 d1] and a half-saturation biomass of 33.33 mented the four functional responses under study (i.e. Fasham, [mmol m3]. Ryabchenko, Real, KTW) in a global marine ecosystem model with 64 phytoplankton species belonging to 4 functional groups which 1 are differentiated by their dependence of growth on external nutri- where gmax [d ] is the maximum biomass-specific ingestion rate of 3 ents. We show that active switching is a mechanism that allows a community of predator species Z [mmol m ]; Vmax = gmaxZ is the 3 1 higher levels of species co-existence, specially when combined maximum ingestion rate [mmol m d ]; dj [n.d.] dictates switch- with strong top-down control (i.e. maximal feeding). We use the ing towards prey pj; Q [n.d.] gives the predators’ probability of S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109 95

feeding as a saturating function of the available (i.e. palatable) prey Table 1 3 3 1 Functional responses. ksat is the half-saturation constant for ingestion [mmol m ]; qj biomass; Gj [mmol m d ] is the ingestion rate upon prey pj; and 3 1 is the constant preference (i.e. not density dependent) [n.d.] for prey pj; and /j is the G [mmol m d ] is the total ingestion rate from all prey. Both dj variable preference (i.e. density dependent) [n.d.] for prey pj. Note that after and Q are non-dimensional terms that vary between 0 and 1. The substitution of the K parameter, the feeding probability Q for the Real and KTW sum of their multiplication across all prey gives the total food formulations becomes identical, and that the only difference between Fasham and limitation for the predator, which will also be between 0 and 1. Ryabchenko formulations is their K parameter. The total ingestion rate will be controlled by Q (see Eq. (2)) while Real Fasham Ryabchenko KTW the fraction of each prey in the diet will be determined by dj (see dj qjpj / p / p / p Eq. (1)). j j j j j j Fq F/ F/ F/ b b The differences between functional responses come from how Q F/ F/ Fq F/ they characterize the prey-switching dj and feeding-probability Q b b K þ F/ K þ F/ b b K þ Fq K þ F/ terms. When the relative frequency of prey eaten is their relative den- Kksat ksat ksat F/ ksat ksat sity in the environment, the switching is passive; otherwise the Fq Fq switching is active. When the total ingestion is a function of total food and independent of the prey biomass distribution, the feeding is maximal; otherwise the feeding is non-maximal. The term d is only dif- j Table 2 ferent for the one passive-switching functional response (Real’s Functional responses (cont.). formulation), while the other three active prey-switching functional X F ¼ / p Total food using variable prey preference responses (Fasham, Ryabchenko, and KTW formulations) share the / X j j Fq ¼ q p Total food using constant prey preference same dj but differ in their Q (see Tables 1 and 2). j j p Pqj j Variable prey preference parameter /j ¼ qjpj

2.1. Fasham’s formulation qj =[0 1] Constant prey preference parameter

Based on the work by Hutson (1984), Fasham et al. (1990) suggested the following formulation to account for predators’ basic multi-species Holling Type III functional response switching towards the most abundant prey species. Note that here (Gentleman et al., 2003; Koen-Alonso, 2007; Prowe et al., 2012b): a predator means a community of predators belonging to a given species instead of a single individual organism: Gj ¼ V maxdjQ ð7Þ

/jpj F/ Gj ¼ V maxdjQ ð3Þ ¼ V ð8Þ max F K þ F / p F / / ¼ V j j / ð4Þ max /jpj F/ K þ F/ ¼ V max P P ð9Þ 2 / p k = q p þ / p jPj sat i i i i ¼ V max ð5Þ ksat þ /ipi 2 qjpj 2 ¼ V max P ð10Þ qjpj 2 2 P P ksat þ qipi ¼ V max 2 ð6Þ ðksat qipiÞþ qipi The only difference between the Ryabchenko and Fasham for- 3 where ksat is the half-saturation constant for ingestion [mmol m ]; mulations is the parameter K of the Q term (see Table 1). In Ryab- qj is the constant preference (i.e. not density dependent) for prey pj chenko’s, K is made variable by scaling the half-saturation constant [n.d.]; and /j is the variable preference (i.e. density dependent) for ksat with the ratio between ksat and the total prey abundance Fq prey pj [n.d.]. The parameter K is related to the half-saturation for (see Tables 1 and 2). In common with Fasham, a switching predator ingestion ksat and in Fasham’s formulation is assumed to be following Ryabchenko’s formulation will concentrate its feeding on constant (see Tables 1 and 2). The predators’ active prey-switching the relatively most abundant prey (Gismervik and Andersen, is controlled by the variable parameter /j, which gives the relative 1997). abundance of each prey measured with respect to the total food available (see Table 2). 2.3. Real’s formulation

The constant preference (qj) can reflect a given prey palatability, the matchup between attack-survival strategies, or be related to Derived from an analogy between feeding and enzyme reac- predator–prey size ratios. The variable preference (/j) is a way to tions (Real, 1977, 1979), this formulation is equivalent to the gen- characterize how predators may select preferentially the most eral (Type II or III) Holling functional response. Extended to abundant prey, reflecting an increase in efficiency at capturing or account for multiple prey, it takes the following form: handling a given prey type as its biomass increases relative to G ¼ V d Q ð11Þ the others. Switching can be interpreted as a way of reflecting a j max j b change in the activity or composition of a heterogeneous commu- qjpj Fq ¼ V ð12Þ nity of predators that is not explicitly resolved in the model (Fas- max b b Fq K þ Fq ham et al., 1990). That is, having one generalist predator with b qjpj Fq active prey-switching is implicitly accounting for having many ¼ V ð13Þ max F b b specialist predators that attack their preferred prey when they be- q ksat þ Fq come available (see Section 4). P q p ðÞq p b P j j Pi i ¼ V max b b ð14Þ qipi k þ ðÞq p 2.2. Ryabchenko’s formulation sat i i The main differences with the two previous functional This formulation has been derived independently by many responses are that Real’s formulation does not account for active authors (Ryabchenko et al., 1997; Gismervik and Andersen, 1997; prey-switching and that it always gives maximal feeding (e.g. see Koen-Alonso, 2007; Smout et al., 2010) and is often referred to as Michaelis–Menten of Class 1 multiple resource functional 96 S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109 responses in Gentleman et al. (2003)). The power b or ‘‘Hill’’ coef- distributed among the prey (see Fig. 3a and b), which leads to the ficient (Mitra and Flynn, 2006) is a parameter that determines if non-maximal feeding of these formulations (see Fig. 4). This the shape of feeding probability Q is Type II or Type III. For compar- decrease in feeding probability does not occur for the other two ison with the previous Ryabchenko sigmoidal response, we have functional responses (Real, KTW) for which the feeding is therefore chosen b = 2. Therefore, we can also call this formulation a multi- always maximal (see Figs. 3c and 3d and 4). In both sub-optimal species Holling Type III functional response for total food. (Fasham) and antagonistic (Ryabchenko) feeding, moving along an isocline of equal total food available (e.g. the dotted line con- 3 2.4. Kill-the-winner (KTW) formulation necting the points Prey 1 = Prey 2 = ksat [mmol m ]inFig. 3) gives different values of feeding probability. Furthermore, with sub- Maximal feeding with active prey-switching can be achieved if optimal feeding even an increase in the abundance of one prey, we use a new scaling factor for the half-saturation constant ksat in while keeping the abundance of the other prey constant (e.g. moving order to obtain a parameter K (see Table 2) that will make the feed- left-to-right along an horizontal line at any Prey 2 biomass), can ing probability Q to be solely a function of the total food Fq and sometimes lead to a decrease in the feeding probability despite the thus independent of the prey biomass distribution: fact that the total food is actually increasing (see Fig. 3a). The third formulation (Real) does not consider active prey- G ¼ V d Q ð15Þ j max j switching. Thus, the feeding probability now depends on total food b /jpj F/ available but not on how biomass is distributed among the prey. ¼ V max b b ð16Þ F This means that if the constant prey preferences qj were all the / K þ F/ same (e.g. say equal to 1.0), all prey would become perfectly sub- / p Fb V j j q 17 stitutable from the point of view of the predator. In this situation ¼ max b b ð Þ F/ k þ F the feeding is always maximal because the presence of other prey sat Pq b does not interfere antagonistically with the predators’ feeding / p ðÞq p P j j Pi i ¼ V max b b ð18Þ probability (see Fig. 3c). Finally, the fourth formulation (i.e. KTW) /ipi k þ ðÞq p sat P i i behaves as a combination of the other three formulations: it q p2 ðÞq p b accounts for active switching while giving maximal feeding (see ¼ V P j j Pi i ð19Þ max p2 b b Fig. 3d). We will next give more details about the differences of qi i ksat þ ðÞqipi each of these four functional responses and elaborate on the The parameter K is now defined as ksat times the ratio between reasons behind their particular behavior. the total prey abundance computed using the variable preference parameter F/ and the total prey abundance computed using the con- 3.1. Fasham’s formulation stant preference parameter Fq (see Tables 1 and 2). The new scaling makes K to be dynamic and decrease when F/ becomes smaller than In this formulation and in common to the other two functional F (e.g. when the food becomes evenly distributed among all prey) q responses with switching, the term dj is a non-linear function of the and simply reflects differential patterns in the attack rates upon each prey relative biomass and rapidly increases when the abundance of prey species (see Appendix A). Basically, this scaling removes the a given prey is high relative to the total food, which rapidly dependence of the feeding probability on F/, and thus its depen- changes the relative fraction of alternative prey in the diet of the dence on the particular distribution of food among the prey. There- predator (see Fig. 2a). Therefore, the predation pressure is dispro- fore, the feeding probability will only change as a function of total portionately large on relatively more abundant prey and dispro- available food Fq and with a constant half-saturation for ingestion portionately small on relatively less abundant prey (Murdoch, ksat (see Eq. (17)). The ecological assumption is that all prey are per- 1969). That is because the non-linear (i.e. quadratic) increase of fectly substitutable for equal fixed preferences (Tilman, 1982). Note the switching term dj with prey pj biomass happens faster than if that the KTW formulation combines the same d as in the Fasham/ j it were simply a linear function of prey pj biomass. This implies Ryabchenko formulations with the same Q as in the Real formula- that relatively low abundant prey are granted implicitly a prey ref- tion. As before, the power b will determine if the shape of Q is Type uge through a relaxation of feeding at low relative prey densities II or Type III for total food. In order to obtain the same feeding prob- (Vallina and Le Quéré, 2011; Prowe et al., 2012a). That is, when ability as Real’s formulations we chose it to be b = 2. We give a formal alternative prey are present the functional response becomes sig- derivation of the KTW formulation in Appendix A where we make an moidal (see the shaded area in Fig. 2a). Thus, the functional re- explicit link between active switching to fundamental properties sponse on Prey 1 goes from being Type II when Prey 2 like the attack rate upon different prey species. abundance is zero to being Type III when Prey 2 abundance is highest (see Fig. 2a). 3. Feeding mode: maximal and non-maximal For a fixed amount of total food, the feeding probability Q decreases with the evenness of the prey biomass distribution (see

Fig. 2 show ingestion upon each prey Gj as a function of the prey Fig. 3a). This happens because Q was calculated using F/ as the biomass pj for an idealized ecosystem consisting of one predator measure of total food, which includes the variable preference /j species feeding upon two prey species with the four functional re- (i.e. the relative abundance of each prey respect to total food; see sponses evaluated in this study. Fig. 3 gives both the feeding prob- Tables 1 and 2). The feeding probability can even decrease with to- ability Q and the total ingestion G from the two prey as a function tal food for moderate increases of one prey biomass, leading to the of pj. The total ingestion is G = VmaxQ and we assume Vmax = 1.0 sub-optimal feeding observed for this formulation (see Fig. 3a). The 3 1 [mmol m d ] for simplicity. In common to all four functional remain issue with using F/ to calculate Q is that for mass conserva- sponses, the total ingestion increases at low prey abundance and tive ecosystems having many co-existing species could imply that then starts to saturate at higher prey abundance (see Fig. 3). Also, each of them represents just a small fraction of the total prey the feeding on a given prey (e.g. Prey 1) is relaxed as the biomass of abundance. Thus, the parameter /j will become small because the alternative prey (i.e. Prey 2) increases (Fig. 2). the fraction of each prey pj abundance respect to a constant total The ﬁrst two functional responses (Fasham, Ryabchenko) prey abundance decreases with the number of prey. Smaller /j will account for active switching but while doing so they decrease make both F/ and Q small as well. Taken to the limit where the rel- the feeding probability as the total available food becomes evenly ative fraction of each prey is inﬁnitesimally small this will lead to S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109 97

3 1 Fig. 2. Shape of the four functional responses (Fasham, Ryabchenko, Real, KTW) for the ingestion Gp1 [mmol m d ] upon Prey 1. Constant parameters: maximum grazing 3 1 3 rate Vmax = 1.0 [mmol m d ], half saturation for ingestion ksat = 33.33 [mmol m ], prey preferences qj = 1.0 [n.d.]. The shaded areas depict the region where the functional response is sigmoidal (i.e. Type III). The non-shaded areas depict the region where the functional response is hyperbolic (i.e. Type II).

(a) Feeding [ Fasham ] (b) Feeding [ Ryabchenko ] 100 1 100 1

80 0.8 80 0.8

60 0.6 60 0.6 Prey 2 40 0.4 Prey 2 40 0.4

20 0.2 20 0.2

0 0 0 0 0 20 40 60 80 100 0 20 40 60 80 100 Prey 1 Prey 1 (c) Feeding [ Real ] (d) Feeding [ KTW ] 100 1 100 1

80 0.8 80 0.8

60 0.6 60 0.6 Prey 2 40 0.4 Prey 2 40 0.4

20 0.2 20 0.2

0 0 0 0 0 20 40 60 80 100 020406080100 Prey 1 Prey 1

3 1 Fig. 3. Countour plots of the feeding probability Q [n.d.] and total ingestion G = VmaxQ [mmol m d ] from Prey 1 and Prey 2 for the four functionalP responses (Fasham, 3 3 Ryabchenko, Real, KTW) as a function of prey abundance [mmol m ]. The dotted line gives an isocline of equal total food available (Fq ¼ qjpj ¼ 33:33 [mmol m ]). 3 1 3 Constant parameters: maximum grazing rate Vmax = 1.0 [mmol m d ], half saturation for ingestion ksat = 33.33 [mmol m ], prey preferences qj = 1.0 [n.d.]. 98 S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109

1 preferences, the fraction of each prey in the diet will simply reﬂect 0.9 Maximal their fraction in the environment. Passive switching does not provide a refuge to relatively less 0.8 Non−maximal abundant prey. In fact, the prey lose their refuge when the biomass 0.7 of alternative prey increases: there is a transition from a Type III 0.6 response to a Type II response (see the non-shaded area in 0.5 Fig. 2c). This differs from the behavior of the Ryabchenko formulation, in which the predation upon each individual prey is always 0.4 sigmoidal. Furthermore, it is exactly the opposite behavior from

Feeding probability 0.3 the Fasham formulation, in which the transition went from Type 0.2 II to Type III. With passive switching any increase in either prey will lead to higher predation on both species, especially at low total 0.1 prey abundance (i.e. values below ksat, before the term Q starts to 0 0 102030405060708090100 saturate). Number of prey species The best way to visualize this behavior is by noting that the grazing refuge now applies to the total food, instead of to each indi- 3 1 Fig. 4. Feeding probability Q [n.d.] and total ingestion G = VmaxQ [mmol m d ] for vidual prey. Starting at low total food abundance, as total food in- a constant total food available of 100 [mmol m3] as a function of the number of creases the refuge will slowly disappear for all prey, regardless of 3 equally abundant prey species (i.e. 100/N) [mmol m ] in an idealized ecosystem their relative abundances. Prey 1 loses its refuge when Prey 2 in- composed of one predator species feeding upon N prey species with: maximal creases because Q increases faster than d decreases. However, ingestion (Real and KTW formulations; black dotted line) and non-maximal 1 ingestion (Fasham and Ryabchenko formulations; gray dotted line). Constant the fact that Q is now computed using Fq, a measure of total food 3 1 parameters: maximum grazing rate Vmax = 1.0 [mmol m d ], half saturation for that does not include the variable preference /j (see Tables 1 and 3 ingestion ksat = 100 [mmol m ], prey preferences qj = 1.0 [n.d.]. (Note that Fasham 2), has the advantage of leading to maximal feeding (see Figs. 3c – and Ryabchenko formulations will give slightly different curves if ksat 100). and 4). the feeding probability tending to zero, regardless of total prey 3.4. KTW formulation abundance (see the non-maximal feeding case in Fig. 4).

This formulation uses the same active switching d as the Fas- 3.2. Ryabchenko’s formulation j ham and Ryabchenko formulations, in combination with the maximal feeding probability Q of Real’s. That is, the variable preference One feature of Ryabchenko’s formulation is that the shape of the / is now only used to compute the switching term but not to com- functional response for predation upon any particular prey p is al- j j pute the feeding probability. The new scaling of the half-saturation ways sigmoidal, even if there is only one prey present (Gismervik constant (i.e. k is multiplied by the ratio F /F ) makes the feeding and Andersen, 1997). Also, and in common to Fasham’s formula- sat / q probability to essentially depend on F (see Tables 1 and 2), which tion, as the biomass of alternative prey increases, the sigmoidal q eliminates the non-maximal feeding observed for the Fasham and shape of the functional response for any given p becomes stronger j Ryabchenko formulations (see Fig. 3). This results in a multi-spe- (see the shaded area in Fig. 2b). Secondly, the feeding probability Q cies functional response that combines maximal feeding with ac- shows lower dependence on prey abundance distribution than the tive switching: more food will always imply higher feeding Fasham formulation and always increases with total food (i.e. it probability (before saturation) and relatively more abundant prey does not cause sub-optimal feeding). Yet and in common with will contribute a larger fraction of the predators’ diet than their Fasham, the feeding probability Q leads to antagonistic feeding fraction in the environment. because it decreases with the evenness of the prey biomass distri- Note that the only mathematical difference between the Real bution (see Fig. 3b). formulation and the KTW formulation is the use of quadratic prey In Ryabchenko’s formulation the feeding probability Q is also abundances in the former to compute the switching term (see Eqs. computed using F (see Tables 1 and 2). However, the scaling of / (14) and (19); see also kill-the-winner coefficient a in Appendix A). the half-saturation constant in the Ryabchenko formulation (i.e. Since Real and KTW use the same feeding probability Q, they both k is first squared and then scaled by the total prey abundance; sat give maximal feeding (see Figs. 3 and 4). However, adding active see Table 1) eliminates the sub-optimal feeding of Fasham’s formu- switching leads to important differences in behavior between the lation (see Fig. 3a and b). Nevertheless, the feature of the feeding two formulations. In particular, we note that now the functional probability tending to zero as the relative fraction of biomass in response for each individual prey p is always Type III, with the sig- each prey becomes infinitely small still persists in Ryabchenko’s j moidal shape becoming stronger as the biomass of alternative prey formulation (see the non-maximal feeding case in Fig. 4). To avoid increases (see the shaded area in Fig. 2d). The prey do not lose their these conceptually problematic issues, the solution we suggest is refuge when the biomass of alternative prey increases, contrary to to remove the variable preference / from the calculation of the j Real’s formulation. This is similar to the Ryabchenko formulation feeding probability Q and use / only to compute the switching j but note that if we chose the power to be b = 1 (instead of b = 2), term d (see Section 3.4). j it will behave more like the Fasham formulation (transition from Type II to Type III). 3.3. Real’s formulation

This formulation does not include active switching. For moder- 4. Food web configuration: explicit and implicit ate to high total prey abundance (i.e. values above ksat) the predation on a given pj will still decrease if the abundance of alternative Fig. 4 shows the feeding probability as a function of the number prey increases but this only reflects that pj becomes a smaller proof equally abundant prey for maximal and non-maximal feeding portion of the total prey abundance in the environment. The term formulations. Although the total food is constant, the ingestion de- dj is now a linear function of the prey biomass and their constant creases exponentially with the number of prey in the non-maximal prey preferences qj. The switching is thus passive: for equal fixed case (Fasham, Ryabchenko) while it is constant when the feeding is S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109 99

Fig. 5. Food web conﬁgurations that are implicit (left-side panels) to an explicit food web (right-side panel) composed of one predator species feeding on N identical prey species using functional responses with maximal feeding (upper panels; i.e. Real and KTW formulations) and non-maximal feeding (bottom panels; i.e. Fasham and Ryabchenko formulations). The dotted lines in the explicit food web reﬂect that the strength of predator–prey interactions decreases with the number of prey for the non- maximal feeding functional responses. See Appendix B.

maximal (Real, KTW). The root cause of this behavior is that respectively), whereas for the Real and KTW formulations the total maximal and non-maximal feeding formulations are implicitly ingestion is independent of N because they are implicitly assuming assuming different food web configurations: switching is essen- a food web of meshwise interactions (see Appendix B). tially a community response. Food web configuration (either explicit or implicit) has important consequences for the interaction 5. Global ocean simulations strength between the whole predator and prey communities (see Appendix B) and for the community assembly process (Grover, We implemented the four functional responses described above 1994; Loreau, 2010). in a global marine ecosystem model (Follows et al., 2007; Having one explicit predator species with active switching can Dutkiewicz et al., 2009) in order to evaluate the impact of different be seen as a way of implicitly accounting for many predator spe- modes of predation (i.e. passive/active switching with maximal/ cies feeding preferentially upon different prey (Fasham et al., non-maximal feeding) on marine phytoplankton diversity and 1990). As some prey species become more relatively abundant, biogeography (Barton et al., 2010; Prowe et al., 2012a). See Supp. this will be followed by an increase in the proportion of their spe- material (S1) for a detailed description of the model. We also cific predators. Non-maximal feeding formulations with one ex- performed a sensitivity analysis to the feeding pressure (i.e. low, plicit predator (see lower-right panel in Fig. 5) are implicitly medium, high) through varying the constant half-saturation assuming a food web configuration of pairwise predator–prey constant for ingestion ksat by ±50% respect to the control case. interactions (i.e. one-to-one) in which a fraction of the commu- The results are the average of these 3 ensemble runs. The individual nity of predators feeds exclusively upon a single prey species runs are given in the Supp. material (S2). (see lower-left panels in Fig. 5). The fraction of the predator community that becomes fully specialized in a single prey species is 5.1. Species traits given by the fraction of that prey in the environment. Maximal feeding formulations with one explicit predator (see upper-right The model was initialized with 64 phytoplankton species panel in Fig. 5) are implicitly assuming a food web configuration belonging to four major phytoplankton functional groups and of meshwise predator–prey interactions (i.e. all-to-all) in which two size-classes: small phytoplankton (i.e. Prochlorococcus and the whole community of predator can feed upon all prey species Synechococcus) and large phytoplankton (i.e. flagellates and dia- (see upper-left panels in Fig. 5). toms). The model also resolves two predator size classes that feed There is an inverse dependence of total ingestion with the num- preferentially on small and large phytoplankton, respectively: a ber of prey (N) in the case of explicit pairwise predator–prey interac- generic micro-zooplankton and a generic meso-zooplankton. For tions that is absent in the case of meshwise predator–prey each phytoplankton group, we generated 16 species by allowing interactions (see the analytical derivation in the Appendix B). With a ±30% variability of the two traits that characterize the groups’ fully specialized pairwise interactions adding more prey implies ability to take up nutrients: the maximum specific growth rate 1 splitting the predators’ biomass into attacking specific prey species, lmax [d ] and the half-saturation constant for nutrient uptake ks which decreases the strength of the non-linear interactions between [mmol m3]. Phytoplankton growth will only be limited by nutri- the whole predator and prey communities. With meshwise interac- ents and light levels, without photo-inhibition or temperature tions adding more prey does not affect the interaction strength be- dependence. Within each group, the most competitive species will tween the whole predator and prey communities (see Appendix be the one having the highest maximum specific growth rate with B). This therefore explains the non-maximal feeding observed for the lowest half-saturation constant, which leads to the highest the Fasham’s and Ryabchenko’s formulations and the maximal feed- uptake affinity (i.e. lmax/ks). Among groups, there is a trade-off ing observed for Real’s and KTW formulations. Both Fasham’s and between growth rate and nutrient affinity, which provides each Ryabchenko’s formulations are implicitly assuming a food web of phytoplankton functional group a particular nutrient niche pairwise interactions (following Type II and Type III responses, (Dutkiewicz et al., 2009). 100 S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109

[ Real ] [ Fasham ] [ Ryabchenko ] [ KTW ] 90 60 30 0 Pro. −30 −60 −90

90 60 30 0 Syn. −30 −60 −90

90 60 30 0 Fla. −30 −60 −90 90 60 30 0 Dia. −30 −60 −90

0.03 0.1 0.3 1 3 0.03 0.1 0.3 1 3 0.03 0.1 0.3 1 3 0.03 0.1 0.3 1 3

Fig. 6. Phytoplankton biomass [mmolN m3] by functional group (Prochlorococcus, Synechococcus, ﬂagellates, diatoms): First column panels: Real formulation, Second column panels: Fasham formulation, Third column panels: Ryabchenko formulation, Fourth column panels: KTW formulation.

The nutrient uptake affinities are related to the species’ subsis- et al., 2009). However, each group tends to occupy a well defined tence nutrient concentration or ‘‘R star’’, which is defined as oceanic region, not being capable of co-existing with other groups mphy due to competitive exclusion. This is more easily noticeable in the R ¼ l m ks. This concept was derived by Tilman (1977) and max phy individual simulations (see Supp. Material S2)thanintheaverage gives the equilibrium requirement of a shared common resource of the ensemble that significantly blurs these features. Active switch- (e.g. phosphate) of a monoculture of each species with constant ing (Fasham, Ryabchenko, KTW) allows for more spatial overlapping loss rates (Tilman, 1982). Note that in this restricted definition of ⁄ of phytoplankton groups; they co-exist over larger regions. R the mortality rate mphy is assumed constant. However, including The biogeography of each group is very sensitive to the choice predators adds an extra mortality that varies with top-down pres- ⁄ of the functional response. Fasham and Ryabchenko formulations sure, causing mphy (and thus R ) to increase with the feeding rate. Species with lower maximum growth rate are more sensitive to give similar phytoplankton biogeography; Real’s gives markedly changes in the mortality rate than species with higher growth rate: different distributions; and KTW gives a biogeography that is inter- ⁄ mediate. This is similar to the conclusions of an earlier study that if mphy and lmax are of similar magnitude, the R will become very high. Therefore, the presence of shared predators adds the poten- tested four functional responses on phytoplankton group biogeog- tial for apparent competition among the prey, which occurs indi- raphy and also found large variations in the extent and magnitude rectly between the species that share a common predator (Holt, of the simulated distributions of several phytoplankton groups 1977; Grover, 1994; Loreau, 2010). with the grazing formulation (Anderson et al., 2010). With active switching (Fasham, Ryabchenko, KTW) the biogeography of the 5.2. Phytoplankton biogeography phytoplankton groups (Fig. 6) matches their species-richness distribution (Fig. 7): the highest diversity in each group is generally The global simulations show that in a non-stationary environ- observed where it dominates. However, when using active switch- ment the four phytoplankton groups are able to persist even with- ing with non-maximal feeding (i.e. Fasham and Ryabchenko) there out active switching (see Real maps in Fig. 6) or even without any is a probably unrealistic dominance of the small Prochlorococcus grazing at all (see Supp. Material S3). Since among groups there is a species over most of the ocean: they display almost global cover- trade-off between growth rate and nutrient affinity that gives each age and their biomass concentration is usually the highest of the phytoplankton group a particular nutrient niche (see Supp. four groups, even in the Southern Ocean (i.e. 40–60°S) where larger Material S1). Seasonality disturbances provide niches for both diatoms are known to dominate the phytoplankton biomass (Boyd low-nutrient adapted groups (i.e. small phytoplankton) and high- et al., 2000; Gall et al., 2001; Hoffmann et al., 2006; Hirata et al., nutrient adapted groups (i.e. large phytoplankton) (Dutkiewicz 2011). S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109 101

[ Real ] [ Fasham ] [ Ryabchenko ] [ KTW ] 90 60 30 0 Pro. −30 −60 −90

90 60 30 0 Syn. −30 −60 −90

90 60 30 0 Fla. −30 −60 −90

90 60 30 0 Dia. −30 −60 −90

0 4 8 12 16 0 4 8 12 16 0 4 8 12 16 0 4 8 12 16

Fig. 7. Species richness by functional group (Prochlorococcus, Synechococcus, flagellates, diatoms): First column panels: Real formulation, Second column panels: Fasham formulation, Third column panels: Ryabchenko formulation, Fourth column panels: KTW formulation. Species richness is defined as the annual mean of monthly diversity. The monthly diversity is defined as the total number of species comprising greater than 1% of the total biomass at that location and month.

When using passive switching with maximal feeding (i.e. Real), weaker predator–prey interaction strength, which allows the total the Prochlorococcus distribution appear more realistic since they prey biomass to attain higher values. are known to mostly dominate oligotrophic regions between 40°N and 40°S(Longhurst, 2006; Moore, 2010; Hirata et al., 5.3. Phytoplankton diversity 2011). However, at higher latitudes (e.g. Southern Ocean) the most dominant group should be the diatoms instead of the flagellates Regarding the global distribution of species richness obtained (Boyd et al., 2000; Gall et al., 2001; Marañon et al., 2001; Hoffmann with the four functional responses, the two main features are: (i) et al., 2006; Hirata et al., 2011). The transition between biogeo- species co-existence within and among phytoplankton functional graphic regions is also very sharp (see Supp. Material S2). When groups when using active switching (i.e. Fasham, Ryabchenko, using active switching with maximal feeding (i.e. KTW) the phyto- KTW) and (ii) dominance of one species per functional group and plankton functional group distributions are more balanced. Each competitive exclusion of all the others within each group when group tends to dominate in some ocean areas with a smooth tran- using passive switching (i.e. Real) (see Fig. 7). The highest level sition between biogeographic regions. The modeled analogs of Pro- of species diversity is obtained with the KTW formulation, both chlorococcus dominate at low latitudes and diatoms dominate at per functional group (see Fig. 7) and total (see Fig. 10). Species rich- high latitudes; Synechococcus analogs are more widely distributed ness is defined here as the annual mean of monthly diversity, than Prochlorococcus; and modeled flagellates are also more widely which is measured as the total number of species contributing distributed than diatom analogs. Although diatoms show the high- greater than 1% of the total biomass at that location and month est local biomass, no functional group clearly dominates in terms (Barton et al., 2010). of global abundance (see Fig. 6). Without active switching (Real) we obtain the lowest level of The feeding probability of zooplankton (see Fig. 8) has a very maximum diversity: 4 species, one per functional group strong influence on total phytoplankton biomass (see Fig. 9): the (Fig. 10c); these are the best competitors of each group (see Supp. Fasham and Ryabchenko formulations lead to much higher bio- Material S1) and therefore they outcompete all other species of mass concentrations than the Real and KTW formulations (up to their group. With active switching plus maximum feeding (KTW) a factor of 3). The lower feeding probability of the non-maximal we obtain the highest level of maximum diversity: 48 species feeding formulations (i.e. Fasham and Ryabchenko) compared to on an annual average (Fig. 10d); with active switching plus non- the maximal feeding formulations (i.e. Real and KTW) means a maximal feeding (Fasham and Ryabchenko) we obtain intermediate 102 S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109

(a) Feeding [ Fasham ] (b) Feeding [ Ryabchenko ] 90 90

60 60

30 30

0 0

−30 −30

−60 −60

−90 −90

0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1

60 60

30 30

0 0

−30 −30

−60 −60

−90 −90

0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1

Fig. 8. Average feeding probability [n.d.] from the two generic zooplankton upon phytoplankton for the four functional responses: (a) Fasham formulation, (b) Ryabchenko formulation, (c) Real formulation, and (d) KTW formulation. levels of diversity (Fig. 10a and b). The Fasham and Ryabchenko for- 1974 with simple Lotka–Volterra models that an increase in in- mulations support lower diversity than the KTW formulation be- ter-specific competition brings instability to the food web if the in- cause their non-maximal feeding decreases the strength of the tra-specific competition remains constant. That is, when the sum active switching stabilizing mechanism. Using Fasham’s formula- of the inter-specific forces in the ecosystem is higher than the tion Prowe et al., 2012a showed that increasing the grazing pres- sum of its intra-specific forces, the system becomes unstable (e.g. sure, increased phytoplankton diversity. We performed a species extinctions occur). Stronger predation with non-selective sensitivity analysis of the feeding pressure that gives similar results feeding falls within this scenario (Haydon, 1994). However, stron- (see Supp. Material S2). When the prey experience less predation ger predation with active switching increases intra-specific forces pressure, they experience more competition for nutrients (Fuchs and thus it stabilizes the ecosystem through a negative (i.e. self- and Franks, 2010). regulatory) feedback affecting each prey biomass (Haydon, 1994). With active switching the lowest diversity is observed in nutri- That is why active switching combined with maximal feeding gives ent poor regions like the subtropical gyres, and the highest diver- the higher level of species diversity. sity occurs at nutrient rich regions like the upwelling system off the coast of Peru (Fig. 10). Slightly different patterns of diversity 6. Discussion were reported by Barton et al., 2010 and Prowe et al., 2012a but our results are not directly comparable to theirs. These studies in- Selective feeding is known to be an important component cluded optimal niches of light and temperature that are absent in underpinning the assembly rules of predator–prey communities our simulations. Differences in light and temperature sensitivities (Grover, 1994; Loreau, 2010) and the size-structure of marine com- affected the species fitness, leading to co-existence at low latitudes munities (Armstrong, 1994; Poulin and Franks, 2010; Banas, 2011). of phenotypes with similar subsistence resource concentrations Small phytoplankton types are good competitors for nutrients but but different light and temperature physiologies. That mechanism are prevented from exhausting all available nutrients by selective is ignored in our simulations for reasons of focus. Grazing induced top-down control (Ward et al., 2012). Grazing places a limit on mortality provides another avenue by which organisms may the amount of phytoplankton biomass within each size-class, ⁄ achieve similar fitness (i.e. R ). while the nutrient supply dictates the number of size classes and Contrary to the results with active switching, increasing the hence the total biomass in the system (Chisholm, 1992; Armstrong, grazing pressure with passive switching has been shown to de- 1994; Ward et al., 2012). This mechanism is based on the size- crease phytoplankton diversity (Prowe et al., 2012a) because specificity of predator–prey interactions which makes the assump- non-selective grazing magnifies the competitive abilities for nutri- tion that grazing interactions occur preferentially at a certain pred- ent uptake of the different prey species which results in stronger ator–prey size ratio. The success of one particular prey is held in competitive exclusion. This relates to the early findings of May, check by the increased growth of the zooplankton that feed prefer- S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109 103

(a) Biomass [ Fasham ] (b) Biomass [ Ryabchenko ] 90 90

60 60

30 30

0 0

−30 −30

−60 −60

−90 −90

0123401234

60 60

30 30

0 0

−30 −30

−60 −60

−90 −90

0123401234

Fig. 9. Total phytoplankton biomass [mmolN m3] for the four functional responses: (a) Fasham formulation, (b) Ryabchenko formulation, (c) Real formulation, and (d) KTW formulation. entially on that prey. Therefore, size-diversity increases with the the phytoplankton diversity and biogeography between the amount of nutrients because the selective top-down control of Fasham/Ryabchenko versus the KTW formulation. Although the smaller sizes allows larger size classes to invade and persist three functional responses withP active switchingP account for selec- 6 (Armstrong, 1994; Fuchs and Franks, 2010; Banas, 2011). tive feeding, the fact that /jpj qjpj implies that with However, predator–prey linkages are not just given by body non-maximal feeding probability the effective half-saturation for sizes but also by the consumers’ feeding type (Wirtz, 2012b). There ingestion upon the whole prey community increases with the is a wide range of prey species at each predator size class (Hansen, number of prey (Banas, 2011) (see Appendix B). That means that 1994; Fuchs and Franks, 2010). Although our global scale simula- the ingestion rate of a given predator species decreases with the tions are based on just two broad size-classes of plankton (i.e. large range of prey species available for consumption (Fuchs and Franks, and small), combining active switching with more highly resolved 2010) which affects the degree of predator-mediated complemen- size-selective feeding will increase the species diversity within tarity (Thebault and Loreau, 2003). Here we argue that the feeding each prey size-class; size preferences alone cannot bring about probability should not necessarily change with the diet breadth. within size-class diversity. Increasing the number of prey species Apparent competition occurs indirectly between prey that in our study is thus analogous to increasing selective predator– share a common predator species (Holt, 1977). This introduces a prey interactions. That is why we find more prey diversity where new element to the prey community assembly which now depends they are more abundant (i.e. less limited by nutrients) in the global not only on the nutrient concentration but also on the predators’ maps with active prey-switching (Fasham, Ryabchenko, KTW). concentration (Grover, 1994; Loreau, 2010). Prochlorococcus tend More nutrients are able to sustain more species through selective to dominate in Fasham’s and Ryabchenko’s formulations because feeding, which is implicit to active switching formulations (see for weak apparent competition among prey they are intrinsically Section 4). Selective feeding (explicit or implicit) leads to positive the best competitors (i.e. lowest R⁄). Complete suppression of graz- niche complementarity among the prey through differentiation of ing activity (i.e. zooplankton dying out of starvation) to evaluate their predators (Loreau, 2010). Prey complementarity arises from the effect of removing apparent competition leads to similar phy- avoidance of predator-mediated or ‘‘apparent’’ competition (Holt, toplankton biogeography, although now with only one species 1977; Loreau, 2010). per phytoplankton group being able to survive due to the lack of Previous works have shown that there are large differences in active switching (see Supp. Material S3). However, strong apparent the phytoplankton diversity (Prowe et al., 2012a) and biogeogra- competition decreases the relative competitive ability of the spe- phy (Anderson et al., 2010) between selective grazing (e.g. Fas- cies with slower maximum growth rate like Prochlorococcus, which ham’s and Ryabchenko’s) versus non-selective grazing (i.e. Real’s) benefits the species with faster maximum growth rates like dia- formulations. However, there are also significant differences in toms (see Supp. Material S2). The apparent competition of the Real 104 S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109

(a) Diversity [ Fasham ] (b) Diversity [ Ryabchenko ] 90 90

60 60

30 30

0 0

−30 −30

−60 −60

−90 −90

0 16 32 48 64 0 16 32 48 64

60 60

30 30

0 0

−30 −30

−60 −60

−90 −90

0 16 32 48 64 0 16 32 48 64

Fig. 10. Total species richness for the four functional responses: (a) Fasham formulation, (b) Ryabchenko formulation, (c) Real formulation, and (d) KTW formulation. Species richness is deﬁned as the annual mean of monthly diversity. The monthly diversity is deﬁned as the total number of species comprising greater than 1% of the total biomass at that location and month.

and KTW formulations is stronger than in Fasham/Ryabchenko for- Should we expect total ingestion always to be maximal? When mulations due to the maximal feeding. the predators must trade off their own predation success against their own risk of predation, maximal feeding may not always be the emergent property (Mariani and Visser, 2010). Also, a decrease 7. Limitations and generality of this work in the predation efficiency with the number of prey species could This work is essentially a theoretical exercise. No attempt has be an emergent property in some complex food webs if there is been made at this stage to formally validate the model simulations an increased difficulty for the predators to attack their prey due with global datasets of phytoplankton diversity and biogeography. to an increase in habitat complexity with prey diversity, such as The main goal of our analyses was to better understand the heterogeneities in prey distribution (i.e. patches) or prey-defense assumptions of switching functional responses with regard to the strategies (Abrams and Allison, 1982; Duffy et al., 2007). On the implicit food web configurations of predator–prey communities, other hand, should we expect total ingestion to become infinitely and to explore the effect that the different functional responses small as prey diversity becomes large? Solutions with negligible for predation may have on modeled ecosystem dynamics. This total ingestion when prey diversity is high appear unphysical and work adds theoretical support to the early suggestion that active enforcing a maximal feeding behavior can alleviate this unrealistic switching is a potentially powerful mechanism to sustain high lev- response. els of diversity (Oaten and Murdoch, 1975; Murdoch and Oaten, Active switching formulations are by design phenomenological 1975). However, is active switching an important driver of diver- models, which describe a phenomenon without explicit consider- sity in natural ecosystems? More experimental and field work ation of the lower-level processes that generate it (Loreau, 2010). should be able to better answer this theoretical hypothesis. Both the Hill coefficient b and the kill-the-winner coefficient a Although switching towards alternative prey has been (see Appendix A) impose density-dependent predatory mortalities documented to occur in the laboratory (Murdoch, 1969, 1975; upon the prey without saying which mechanisms are causing it. Hughes and Croy, 1993; Gismervik and Andersen, 1997; Kiørboe They are an abstraction for a wealth of factors that are neither well et al., 1996; Elliott, 2006; Kiørboe, 2008; Kalinkat et al., 2011), con- understood nor explicitly modeled. The focus is thus placed on the flicting results based on the analyses of even the same data sets consequences that switching has on ecosystem dynamics, rather have been also reported (e.g. (Rindorf et al., 2006; Kempf et al., than on the causes that generate the switching itself. The main 2008)). Thus, an unambiguous proof of its relevance in the field strength and the corresponding main weakness of phenomenologi- remains elusive (Hassell, 2000; Elliott, 2006). cal models is their simplicity: on the one hand they provide simple S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109 105 predictions and clear interpretations; on the other hand they do not dj aj ¼ a ð21Þ provide a complete description of reality (Loreau, 2010). The lack of /j well-defined first-principle rules to model active switching from P Pqj j pure mechanistic grounds can lead to formulation inconsistencies /j ¼ ð22Þ q Pi (see Appendix C). Therefore active switching formulations should i i q Pa ideally move towards a more mechanistic approach. Also, it is worth d ¼ P j j ð23Þ j q Pa noting that there is a clear mismatch between the scales at which the i i i ! plankton communities interact and the scales at which global ocean X b1 0 models operate (Siegel, 1998). a ¼ a qiPi ð24Þ i 8. Conclusions where a is a density-dependent attack rate [(m3 mmol1)d1]; a0 is a density-independent (i.e. constant) attack rate [(m3 mmol1)b d1]; / Complex food web models need mechanisms to overcome the j is the relative abundance of prey species j [n.d.]; d regulates the probably unrealistic but common outcome of one or few species j switching towards prey species j [n.d.]; b is the Hill coefficient that outcompeting all others. The use of functional responses with ac- measures how the attack rate varies with total prey density [n.d.]; tive prey-switching can help alleviate competitive exclusion. How- and is the kill-the-winner coefficient that measures the potential ever, active switching formulations in which the feeding is non- a for selective predation: when is equal to 1.0 we have that a = a maximal (Fasham, Ryabchenko) have the problem that an increase a j and the switching is passive (i.e. no KTW predation); when is bigger in the number of modeled prey species implies a decrease of the a than 1.0 we have that a T a and the switching is active (i.e. KTW pre- average predator–prey interaction strength (Vallina and Le P j P dation). Note that a P a P which implies that total feeding Quéré, 2011). If the strength of predator–prey interactions de- jqj j ¼ qj j is always maximal: if some species are being attacked faster then creases, the total prey biomass will therefore increase. This mostly some others must be being attacked slower. applies to closed (i.e. mass conservative) systems like the global Substituting Eqs. (21) and (22) into Eq. (20): ocean in which the total mass (biotic + abiotic) of essential elements is roughly constant at the seasonal scale and therefore more P prey species means less biomass per prey, until reaching a point adj q Pi=q Pj q PjZ P i iP j j where they all become protected from predation in their own prey Gj ¼ 1 þða=hÞ dk q Pi=q Pk q Pk refuge and none can be eaten. With maximal feeding formulations P k i i k k ad q P Z (Real, KTW) the average of predator–prey interactions is unaffected ¼ j Pi i i P 1 a=h P d by the number of prey species. However, the number of species de- þð PÞ iqi i k k ð25Þ creases to very low levels in the absence of active switching due to adj q PiZ ¼ i Pi 1 a=h P strong competitive exclusion (Real). We derived a kill-the-winner þðP Þ iqi i functional response that combines active switching with maximal hdj iqPiPiZ feeding (KTW). Global ocean simulations show that both active ¼ ðh=aÞþ iqiPi switching and maximal feeding are key elements to sustain higher levels of species diversity while providing realistic phytoplankton Substituting Eq. (24) into Eq. (25): functional-group biogeography. P hdj iqiPiZ Gj ¼ P P Acknowledgments ðh=ða0ð q P Þb1ÞÞ þ q P i i i i i i P P b1 hdj iqiPi iqiPi Z This work was supported by a Marie Curie Fellowship ¼ P P ð26Þ 0 b1 (IOF – FP7) to S.M.V. from the European Union (EU) and was ðh=a Þþ iqiPi iqiPi P performed within the MIT’s Darwin Project. We would like to hd q P bZ j ii i thank Prof. Kai W. Wirtz and three other anonymous reviewers ¼ P b ðh=a0Þþ q P for their thorough review of the manuscript. Our special i i i thanks to Oliver Jahn for his assistance with the global ocean We can now define: simulations.

V max ¼ hZ ð27Þ Appendix A. Kill-the-winner functional response b 0 ksat ¼ h=a ð28Þ The generic functional response for feeding on multiple prey where V is the maximum ingestion rate [mmol m3 d1] and k was mathematically derived by Murdoch (1973). The ingestion max sat is the half-saturation constant for ingestion [mmol m3]. Substitut- rate upon a single prey species j is characterized using the following Eqs. (27) and (28) into Eq. (26): ing expression: P a q P Z a b P j j j q P q P Z Gj ¼ ð20Þ P j j ii i 1 þ ða =h Þq P Gj ¼ V max a P b k k k k k q P b i i i ksat þ iqiPi 3 where Z is the concentration of predators [mmol m ]; Pj is the Fb ð29Þ 3 concentration of prey species j [mmol m ]; aj is the predators’ ¼ V maxdj b b 3 1 1 ksat þ F attack rate on prey species j [(m mmol )d ]; hj is the predators’ 1 ¼ V d Q handling rate of prey species j [d ]; and qj is a constant prey max j preference [n.d.]. Assuming that the handling rates are constant and the same for all prey species (i.e. hj = h) implies that any where Q gives the overall feeding probability [n.d.];P dj dictates the switching towards particular prey species will solely be given by switching towards prey species j [n.d.]; and F ¼ iqiPi is the total differences in the attack rates: food available [mmol m3]. 106 S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109

Appendix B. Food web conﬁguration and feeding mode (two predators–two prey). Thus increasing N in the ecosystem will necessarily decrease the predators’ total ingestion. This is analogous B.1. Explicit food webs to the non-maximal feeding formulations (i.e. Fasham’s and Ryabchenko’s). B.1.1. Pairwise interactions When a food web is composed by specialized predators that can B.1.2. Meshwise interactions only feed upon one single prey species (see lower-left panels in When a food web is composed by generalist predators that feed Fig. 5), increasing the number of prey will decrease the total feed- on all prey (see upper-left panels in Fig. 5) and the total biomass of ing of the predator community as a whole. Let us call N the number both prey and predators is constant, the feeding of the whole pred- of prey and predators that are present in the food web. If the total ator community will be independent of the number of prey and biomass P of prey and the total biomass Z of predators is constant, predators N in the system. The predators’ total ingestion rate G will we have that: be given by: P b XN N XN i Pi P ¼ Pj ð30Þ G ¼ gmaxZj P b ð40Þ j N j K þ i Pi XN Z ¼ Zj ð31Þ where the power b will regulate the functional response (Type II j when b = 1 and Type III if b = 2) and K is a parameter related to In the simplest scenario in which all the prey have equal bio- the half-saturation constant for ingestion. Substituting Eqs. (30) mass and all the predators have equal biomass, we have that: and (33) into Eq. (40) gives:

XN b Pj ¼ P=N ð32Þ P G ¼ gmaxðZ=NÞ b ð41Þ Zj ¼ Z=N ð33Þ j K þ P b The predators’ total ingestion rate G will be given by: P ¼ gmaxZ b ð42Þ XN b K þ P Pj G ¼ gmaxZj b ð34Þ If b = 1 and K = ksat, we have the Type II case: j K þ Pj P where the power b will regulate the functional response (Type II G ¼ gmaxZ ð43Þ ksat þ P when b = 1 and Type III if b = 2) and K is a parameter related to If b = 2 and K k2 , we have the Type III case: the half-saturation constant for ingestion. Substituting Eqs. (32) ¼ sat and (33) into Eq. (34) gives: P2 G ¼ gmaxZ 2 ð44Þ XN b 2 ðP=NÞ ksat þ P G ¼ gmaxðZ=NÞ b ð35Þ j K þðP=NÞ In either case, the predators’ total ingestion is independent of N. b That means that changing the number of prey and predators in the ðP=NÞ ¼ gmaxZ b ð36Þ ecosystem will have no effect on the predators’ total ingestion as a K þðP=NÞ whole. This reflects that all the predator biomass is always inter- b P acting with all the prey biomass, regardless of N. Thus, increasing ¼ gmaxZ ð37Þ NbK þ Pb N in the ecosystem will not affect the overall predator–prey interaction strength. This is analogous to the maximal feeding formula- If b = 1 and K = k , we have the Type II case: sat tions (i.e. Reals’s and KTW). P G ¼ gmaxZ ð38Þ Nksat þ P B.2. Implicit food webs If b = 2 and K ¼ k2 =N, we have the Type III case: sat B.2.1. Non-maximal feeding P2 For non-maximal feeding formulations, the total ingestion rate G g Z 39 ¼ max 2 2 ð Þ G is given by: Nksat þ P XN 2 Pj In either case, N is multiplying the half-saturation constant for G ¼ gmaxZ P ð45Þ NP2 ingestion ksat. Thus, as we increase the number of prey with their j v þ i i own specific predator in the ecosystem, the total ingestion of the where N is the number of prey species (note that there is only one whole predator community will decrease exponentially. This simply predator species) and v is a parameter related to the half-saturation reflects that increasing N implies splitting the predators’ biomass constant for ingestion. Substituting Eq. (32) into Eq. (45) gives: into attacking specific prey. If we have a food web composed of only XN 2 XN 2 one prey and one predator (N = 1), all the predators’ biomass is ðP=NÞ P P interacting with all the prey’s biomass; whereas if we have a food G ¼ gmaxZ N 2 ¼ gmaxZ 2 2 j v þ i ðP=NÞ j N v þ NP web composed of two prey and two predators (N = 2), now half of ð46Þ XN 2 2 all the predators’ biomass is interacting with half of all the prey’s P P ¼ g ðZ=NÞ ¼ g Z biomass while the other half biomass are interacting independently. max 2 max 2 j Nv þ P Nv þ P Therefore, the overall predator–prey interaction strength decreases with N. For example, if the total biomass of either predators and If v = ksatP, we have the solution for Fasham’s formulation: 3 prey is 100 [mmolC m ], and assuming for simplicity Lotka–Vol- P terra interactions (Z P), we can see that 100 100 = 10,000 (one G ¼ gmaxZ ð47Þ Nksat þ P predator–one prey) is bigger than (50 50) + (50 50) = 5000 S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109 107

2 If v ¼ ksat, we have the solution for Ryabchenko’s formulation: This becomes clearer if we consider the case of an omnivorous predator that feeds on two different food classes, say phytoplank- P2 G g Z 48 ton and bacteria, with only one prey species per food class. If both ¼ max 2 2 ð Þ Nksat þ P food classes are present with the same relative abundance (i.e. 50% each), the predator will eat the same amount of phytoplankton and Note that Eq. (47) is equivalent to Eq. (38) and that Eq. (48) is bacteria. If we now subdivide the phytoplankton food class into equivalent to Eq. (39). This means that under the condition of con- two identical prey species, each contributing 25% of the total food, stant total prey and predator biomass, and for the simplest sce- while keeping bacteria as one prey species contributing 50% of the nario in which all prey have equal biomass, the Fasham and total food, the predator will now eat more bacteria than phyto- Ryabchenko formulations are implicitly resolving a community of plankton (66.66% versus 33.33%) even though the combined specialist predators (not explicitly modeled) that form a food amount of phytoplankton and bacteria has remain unchanged web of pairwise predator–prey interactions in which each predator (50–50%). Yet, the predators’ total feeding will be maximal and ex- feeds upon one single prey species (see lower panels in Fig. 5). actly the same for both scenarios; only the proportion of ingestion from each food class has changed. B.2.2. Maximal feeding Subdivision of the prey species present in one food class can For maximal feeding formulations, the predators’ total ingestion thus alter the relative ingestion of the other food classes. This rate G is given by: behavior has no ecological basis and should be avoided or at least XN a b P P minimized. The cause is the way the prey-switching term dj is com- P j G ¼ gmaxZ N b ð49Þ puted: the squared biomass of each prey species is compared to the Pa K þ P j i i sum of the squared biomass of all prey species, regardless of how where the power b will regulate the functional response for total closely related they might be. One sensible solution is to group bio- food ingestion (Type II when b = 1 and Type III if b = 2), and the logically similar prey species into the same category or food class power a will regulate the prey switching (passive if a = 1 and active (e.g. primary producers, herbivores, carnivores, etc.), and then if a = 2; i.e. Real and KTW formulations, respectively). N is the compute the prey-switching term for the prey within each food number of prey species (note that there is only one predator species) class independently. Switching among food classes can also be in- and K is a parameter related to the half-saturation constant for cluded (Ward et al., 2012). The generalized form of the KTW for- b mulation that fulﬁlls the common sense criterion for an ingestion (i.e. K ¼ ksat ). Substituting Eq. (32) into Eq. (49) gives: indeterminate number of food classes i and prey species j per food XN a b ðP=NÞ P class (under the condition that food classes themselves will not be G ¼ g Z P ð50Þ max N a b further subdivided) is thus: j i ðP=NÞ K þ P XN a b a c ðP=NÞ P q P F Fb P j j i ¼ gmaxZ a ð51Þ Gp ¼ V max P ð54Þ NðP=NÞ K Pb j Pa Fc b b j þ qk k k ksat þ F XN b P ¼ V maxdjdiQ ð55Þ ¼ gmaxðZ=NÞ b ð52Þ P K þ P j wherePF ¼ Fk gives the total food available from all food classes; b P Fi ¼ qkpk gives the food available from all prey within food class ¼ gmaxZ b ð53Þ i; and the parameter q is a constant prey preference. The term d K þ P j j dictates the switching towards prey j within food class i, and the Note that Eq. (53) is equivalent to Eq. (42). This means that un- term di dictates the switching towards food class i. The power a will der the condition of constant total prey and predator biomass, and regulate the switching for prey species j: passive when a = 1 and ac- for the simplest scenario in which all prey have equal biomass, the tive if a = 2. The power c will regulate the switching for food class i: Real and KTW formulations are implicitly resolving a community passive when c = 1 and active if c = 2. The power b will regulate the of generalist predators (not explicitly modeled) that form a food functional response for total food: Type II when b = 1 and Type III web of meshwise predator–prey interactions in which all predators when b = 2. In all cases the total food ingestion will be maximal. feed upon all prey (see upper panels in Fig. 5).

Appendix D. Supplementary data Appendix C. The common sense criterion Supplementary data associated with this article can be found, in When a given species is divided into two (or more) identical the online version, at http://dx.doi.org/10.1016/j.pocean.2013. species with a combined density equal to the density of the origi- 08.001. nal species, this should not lead to a difference in the total amount of that species eaten (Arditi and Michalski, 1995; Berryman et al., References 1995). Maximal feeding is a necessary, although not sufficient, condition for this common sense criterion to hold. Therefore both Fas- Abrams, P.A., Allison, T.D., 1982. Complexity, stability and functional response. The ham’s and Ryabchenko’s formulations violate this criterion under American Naturalist 119 (2), 240–249. all conditions. On the other hand, while Real’s formulation always Adjou, M., Bendtsen, J., Richardson, K., 2012. Modeling the influence from transport, fulfills the common sense criterion under any condition, the KTW mixing and grazing on phytoplankton diversity. Ecological Modelling 225, 19– 27. formulation fulfills this criterion in the simplest case where one Anderson, T.R., Gentleman, W.C., Sinha, B., 2010. Influence of grazing formulations prey species is subdivided into two (or more) identical species. on the emergent properties of a complex ecosystem model in an ocean general However, it can violate the criterion for more complicated cases circulation model. Progress in Oceanography 87 (1-4), 201–213. Arditi, R., Michalski, J., 1995. Nonlinear food web models and their responses to where there are several different classes of prey (e.g. several food increased basal productivity. In: Polis, G.A., Winemiller, K.O. (Eds.), Food Webs: classes such as phytoplankton and bacteria) one of which is subdi- Integration of Patterns and Dynamics. Chapman & Hall, London, England, pp. vided into two (or more) others that are identical. In that case the 122–133. Armstrong, R.A., 1994. Grazing limitation and nutrient limitation in marine total ingestion from the food class being subdivided will decrease ecosystems: steady-state solutions of an ecosystem model with multiple food with the number of subdivisions (Visser and Fiksen, 2013). chains. Limnology and Oceanography 39 (3), 597–608. 108 S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109

Armstrong, R.A., 1999. Stable model structures for representing biogeochemical Kempf, A., Floeter, J., Temming, A., 2008. Predator–prey overlap induced Holling diversity and size spectra in plankton communities. Journal of Plankton type III functional response in the North Sea fish assemblage. Marine Ecology Research 21 (3), 445–464. Progress Series 357, 295–308. Armstrong, R.A., McGehee, R., 1980. Competitive exclusion. The American Naturalist Kiørboe, T., 2008. A Mechanistic Approach to Plankton Ecology. Princeton University 115 (2), 151–170. Press, Princeton, New Jersey. Banas, N.S., 2011. Adding complex trophic interactions to a size-spectral plankton Kiørboe, T., Saiz, E., Viitasalo, M., 1996. Prey switching behaviour in the planktonic model: emergent diversity patterns and limits on predictability. Ecological copepod Acartia tonsa. Marine Ecology Progress Series 143, 65–75. Modelling 222, 2663–2675. Koen-Alonso, M., 2007. A process-oriented approach to the multispecies functional Barton, A., Dutkiewicz, S., Flierl, G., Bragg, J.G., Follows, M.J., 2010. Patterns of response. In: N. Rooney, K.S. McCann, D.N. (Ed.), Energetics to Ecosystems: The diversity in marine phytoplankton. Science 327, 1509–1511. Dynamics and Structure of Ecological Systems. Springer, Dordrecht, The Berryman, A.A., Michalski, J., Gutierrez, A.P., Arditi, R., 1995. Logistic theory of food Netherlands, p. 265 (chapter 1). web dynamics. Ecology 76, 336–343. Longhurst, A.R., 2006. Ecological Geography of the Sea. Academic Press, San Diego, Boyd, P.W., Watson, A.J., Law, C.S., Abraham, E.R., Trull, T., Murdoch, R., Bakker, California. D.C.E., Bowie, A.R., Buesseler, K.O., Chang, H., Charette, M., Croot, P., Downing, K., Loreau, M., 2010. From Populations to Ecosystems: Theoretical Foundations for a Frew, R., Gall, M., Hadfield, M., Hall, J., Harvey, M., Jameson, G., LaRoche, J., New Ecological Synthesis. Princeton University Press, Princeton, New Jersey. Liddicoat, M., Ling, R., Maldonado, M.T., McKay, R.M., Nodder, S., Pickmere, S., MacArthur, R.H., Pianka, E.R., 1966. On optimal use of a patchy environment. The Pridmore, R., Rintoul, S., Safi, K., Sutton, P., Strzepek, R., Tanneberger, K., Turner, American Naturalist 100 (916), 603–609. S., Waite, A., Zeldis, J., 2000. A mesoscale phytoplankton bloom in the polar Marañon, E., Holligan, P.M., Barciela, R., Gonzalez, N., Mauriño, B., Pazo, M.J., Varela, Southern Ocean stimulated by iron fertilization. Nature 407, 695–702. M., 2001. Patterns of phytoplankton size structure and productivity in Chisholm, S.W., 1992. Phytoplankton size. In: Falkowski, P., Woodhead, A.D. (Eds.), contrasting open-ocean environments. Marine Ecology Progress Series 216, Primary Productivity and Biogeochemical Cycles in the Sea (Environmental 43–56. Science Research), vol. 43. Plenum Press, New York, pp. 213–237. Mariani, P., Visser, A.W., 2010. Optimization and emergence in marine ecosystem Duffy, J.E., Cardinale, B.J., France, K.E., McIntyre, P.B., Theabault, E., Loreau, M., 2007. models. Progress in Oceanography 84, 89–92. The functional role of biodiversity in ecosystems: incorporating trophic May, R.M., 1974. Stability and Complexity in Model Ecosystems. Princeton complexity. Ecology Letters 10, 522–538. University Press, Princeton, New Jersey. Dutkiewicz, S., Follows, M.J., Bragg, J.G., 2009. Modeling the coupling of ocean Mitra, A., Flynn, K.J., 2006. Accounting for variation in prey selectivity by ecology and biogeochemistry. Global Biogeochemical Cycles 23, 1–15. zooplankton. Ecological Modelling 199, 82–92. Elliott, J.M., 2006. Prey switching in Rhyacophila dorsalis (Trichoptera) alters with Moore, L.R., 2010. Prochlorococcus and other photosynthetic picoplankton. In: larval instar. Freshwater Biology 51, 913–924. Encyclopedia of Life Sciences (ELS). John Wiley & Sons, Ltd., Chichester, United Fasham, M.J.R., Ducklow, H.W., McKelvie, D.S., 1990. A nitrogen-based model of Kingdom, pp. 1–10. plankton dynamics in the oceanic mixed layer. Journal of Marine Research 48, Morozov, A.Y., 2010. Emergence of Holling type III zooplankton functional response: 591–639. bringing together field evidence and mathematical modelling. Journal of Follows, M.J., Dutkiewicz, S., Grant, S., Chisholm, S.W., 2007. Emergent Theoretical Biology 265, 45–54. biogeography of microbial communities in a model ocean. Science 315, 1843– Murdoch, W.W., 1969. Switching in general predators: experiments on predator 1846. specificity and stability of prey populations. Ecological Monographs 39 (4), Fuchs, H.L., Franks, P.J.S., 2010. Plankton community properties determined by 335–354. nutrients and size-selective feeding. Marine Ecology Progress Series 413, Murdoch, W.W., 1973. The functional response of predators. Journal of Applied 1–15. Ecology 14, 335–341. Gall, M.P., Boyd, P.W., Hall, J., Safi, K.A., Chang, H., 2001. Phytoplankton processes. Murdoch, W.W., 1975. Switching in predatory fish. Ecology 56 (5), 1094–1105. Part 1: community structure during the Southern Ocean iron release Murdoch, W.W., Oaten, A., 1975. Predation and population stability. Advances in experiment (SOIREE). Deep-Sea Research II 48, 2551–2570. Ecological Research 9, 1–131. Gentleman, W., Leising, A., Frost, B., Strom, S., Murray, J., 2003. Functional responses Oaten, A., Murdoch, W.W., 1975. Switching, functional response, and stability in for zooplankton feeding on multiple resources: a review of assumptions and predator–prey systems. The American Naturalist 109 (967), 299–318. biological dynamics. Deep-Sea Research II 50, 2847–2875. Pahlow, M., Prowe, A.E.F., 2010. Model of optimal current feeding in zooplankton. Gismervik, I., Andersen, T., 1997. Prey switching by Acartia clausi: experimental Marine Ecology Progress Series 403, 129–144. evidence and implications of intraguild predation assessed by a model. Marine Poulin, F.J., Franks, P.J.S., 2010. Size-structured planktonic ecosystems: constrains, Ecology Progress Series 157, 247–259. controls and assembly instructions. Journal of plankton research 32 (8), 1121– Grover, J.P., 1994. Assembly rules for communities of nutrient-limited plants and 1130. specialist herbivores. The American Naturalist 143 (2), 258–282. Prowe, A.E.F., Pahlow, M., Dutkiewicz, S., Follows, M.J., Oschlies, A., 2012a. Top- Hansen, B.W., 1994. The size ratio between planktonic predators and their prey. down control of marine phytoplankton diversity in a global ecosystem model. Limnology and Oceanography 39 (2), 395–403. Progress in Oceanography 101, 1–13. Hardin, G., 1960. The competitive exclusion principle. Science 131, 1292–1297. Prowe, A.E.F., Pahlow, M., Oschlies, A., 2012b. Controls on the diversity–productivity Hassell, M.P., 2000. The Spatial and Temporal Dynamics of Host–Parasitoid relationship in a marine ecosystem model. Ecological Modelling 225, 167–176. Interactions. Oxford Series in Ecology and Evolution. Oxford University Press, Real, L.A., 1977. The kinetics of functional response. The American Naturalist 111 New York. (978), 289–300. Haydon, D., 1994. Pivotal assumptions determining the relationship between Real, L.A., 1979. Ecological determinants of functional response. Ecology 60 (3), stability and complexity: an analytical synthesis of the stability-complexity 481–485. debate. The American Naturalist 144 (1). Rindorf, A., Gislason, H., Lewy, P., 2006. Prey switching of cod and whiting in the Hirata, T., Hardman-Mountford, N.J., Brewin, R.J.W., Aiken, J., Barlow, R., Suzuki, K., North Sea. Marine Ecology Progress Series 325, 243–253. Isada, T., Howell, E., Hashioka, T., Noguchi-Aita, M., Yamanaka, Y., 2011. Ryabchenko, V.A., Fasham, M.J.R., Kagan, B.A., Popova, E.E., 1997. What causes short- Synoptic relationships between surface chlorophyll-a and diagnostic pigments term oscillations in ecosystem models of the ocean mixed layer? Journal of specific to phytoplankton functional types. Biogeosciences 8, 311–327. Marine Systems 13, 33–50. Hoffmann, L.J., Peeken, I., Lochte, K., Assmy, P., Veldhuis, M., 2006. Different Siegel, D.A., 1998. Resource competition in a discrete environment: why are reactions of Southern Ocean phytoplankton size classes to iron fertilization. plankton distributions paradoxical? Limnology and Oceanography 43 (6), Limnology and Oceanography 51 (3), 1217–1229. 1133–1146. Holling, C.S., 1959. The components of predation as revealed by a study of small Smout, S., Asseburg, C., Matthiopoulos, J., Fernandez, C., Redpath, S., Thirgood, S., mammal predation on the European pine sawfly. The Canadian Entomologist 91 Harwood, J., 2010. The functional response of a generalist predator. Public (5), 293–320. Library of Science – One 5 (5), 1–7. Holt, R.D., 1977. Predation, apparent competition, and the structure of prey Thebault, E., Loreau, M., 2003. Food-web constrains on biodiversity - ecosystem communities. Theoretical Population Biology 12 (2), 197–229. functioning relationships. PNAS 100 (25), 14949–14954. Holt, R.D., 1983. Optimal foraging and the form of the predator isocline. The Thingstad, T.F., 2000. Elements of a theory for the mechanisms controlling American Naturalist 122 (4), 521–541. abundance, diversity, and biogeochemical role of lytic bacterial viruses in Hughes, R.N., Croy, M.I., 1993. An experimental analysis of frequency-dependent aquatic systems. Limnology and Oceanography 45 (6), 1320–1328. predation (switching) in the 15-spined stickleback, Spinachia spinachia. Journal Thingstad, T.F., Lignell, R., 1997. Theoretical models for the control of bacterial of Animal Ecology 62 (2), 341–352. growth rate, abundance, diversity and carbon demand. Aquatic Microbial Hutchinson, G.E., 1961. The paradox of the plankton. The American Naturalist 95 Ecology 13, 19–27. (882), 137–145. Tilman, D., 1977. Resource competition between planktonic algae: an experimental Hutson, V., 1984. Predator mediated coexistence with a switching predator. and theoretical approach. Ecology 58, 338–348. Mathematical Biosciences 68, 233–246. Tilman, D., 1982. Resource Competition and Community Structure. Princeton Ivlev, V.S., 1961. Experimental Ecology of the Feeding of Fishes. Yale University University Press. Press, Yale. Vallina, S.M., Le Quéré, C., 2011. Stability of complex food webs: resilience, Kalinkat, G., Rall, B.C., Vucic-Pestic, O., Brose, U., 2011. The allometry of prey resistance, and the average interaction strength. Journal of Theoretical Biology preferences. PLoS ONE 6 (10), 1–11. 272, 160–173. S.M. Vallina et al. / Progress in Oceanography 120 (2014) 93–109 109

Visser, A.W., Fiksen, Ø., 2013. Optimal foraging in marine ecosystem models: Wirtz, K.W., 2012a. Intermittency in processing explains the diversity and shape of selectivity, proﬁtability and switching. Marine Ecology Progress Series 473, 91– functional grazing responses. Oecologia 169, 879–894. 101. Wirtz, K.W., 2012b. Who is eating whom? morphology and feeding type determine Ward, B.A., Dutkiewicz, S., Jahn, O., Follows, M.J., 2012. A size-structured food-web the size relationship between planktonic predators and their ideal prey. Marine model for the global ocean. Limnology and Oceanography 57 (6), 1877–1891. Ecology Progress Series 445, 1–12.